If we suppose to be infinitely small, then m is the metacentre. And in this case, I and E may be considered to be infinitely small prismatic triangles. And the planes of floatation before inclination, and when the vessel is inclined, may be supposed to intersect in the longitudinal axis of the ship, which intersects the line GF (and the plane of the paper) at right angles.

Then the elementary triangles of the prismatic solids, E and I, are

If the axis of x be the longitudinal axis of the ship, in the plane of floatation.

Hence, the moment of one of these elementary

but,

Q: y'. HT: Hm (by similar triangles.)

From the foregoing calculation we find that the equilibrium is stable if

But from page 67 we find that the equilibrium is stable if

flodx>D. HG

or if, as above, we suppose the plane of floatation to be the plane of xy, the axis of a being parallel

to the longitudinal axis of the ship, and the axis of y being transversely to the length of the ship, the origin being in the line HG; then / becomes y, and da becomes ydx,

:. fl2dλ=fy3.dx

Hence, the equilibrium is stable, also, if

fy3. dx>D. HG

Comparing these results, we find that the metacentre may be below G, and yet the equilibrium remain stable, provided that

Hm>2 HG.

We may here remark that the stability depends upon the lowness of the centre of gravity of the ship, and the height of the centre of displacement; and also (in order that m may be high) on the breadth of the load-water-line. This will be made more clear by what follows.

We will resume the former expression for the moment producing stability, when the ship is inclined through a finite angle

D.GV=b.E-D. GH sin

Now let us suppose that, in the half breadth plan, the lines representing the vertical sections are pro

duced and made proportional to the areas of those sections respectively; and let a curve be drawn through the extremities of these lines. Then, with the same rectangular axes as before, we have

D=Kfydx,

(where K is some constant by which the areas were divided, so that the line representing each area

area

respectively equals ; and y, in this instance,

K

extends over the whole of the latter curve.) Also

..

2

D. GV = 3 +fy3dx— K. GH sin ösydæ

where the first integral extends over the load-waterline alone, and the second over the whole area contained by the other curve.

Hence we see that, in order for the stability to be great, we must have fy'da as great as possible, and the negative term as small as possible; or the area of the load-water-line must be as great as possible, the centre of gravity of the whole ship very low, and the vessel's bottom as lean as possible. This is a very curious result, and contrary to the

almost universal assumption of ship builders. But it will not astonish us if we consider the direction

of the supporting efforts of the water at different heights along the surface of the vessel, and the moments of these efforts. These moments being much greater in the vessel with the lean bottom; and nearly all of them tending to bring the vessel back to her original position, which is not the case when the bilge is fuller.

It has been a matter of wonder, why the flatbottomed coasting vessels of America should be so much stiffer than their other vessels (vide Griffiths); but the above fact quite explains the phenomenon.

We see, hence, that it would be a good plan to build small sailing-boats very shallow, and with a very straight floor, forming an angle of at least 60°, with the plane through the stern-post and keel. And, in order to prevent them from being leewardly, a lee-board could be provided, with a tackle to keep that lee-board vertical when the vessel is inclined.

A good practical mode of ascertaining the relative values of flådλ in different vessels, would be to

make a model of half the

water-line, and make it

oscillate around the mid

dle-line WL (which must be quite horizontal); and the time of a single oscillation must be found by counting the number of oscillations in one minute. Let the time of an oscillation be t' seconds, suppose